She did know Kushina. Or, at the very least, she knew of her. Kushina was a well-known jonin before going on maternal leave.
As for what jutsus Tsunade has that could be Uzumaki-related, the Yin Seal comes to mind. It's not confirmed an Uzumaki clan jutsu, but Mito seemed to have it, and it is a very powerful sealing jutsu, which the Uzumaki clan is renowned for.
Well... no, not really. The First and Second Hokage have no blood relation at all to Naruto, and it's not clear what Mito's direct relationship with Kushina was. Even though they're of the same clan, they didn't necessarily have to be a part of the same branch in said clan. Tsunade's probably very distantly related to Naruto.